In Our Time - Pythagoras

Episode Date: December 10, 2009

Melvyn Bragg and guests Serafina Cuomo, John O'Connor and Ian Stewart discuss the ideas and influence of the Greek mathematician Pythagoras and his followers, the Pythagoreans.The Ancient Greek mathematician Pythagoras is probably best known for the theorem concerning right-angled triangles that bears his name. However, it is not certain that he actually developed this idea; indeed, some scholars have questioned not only his true intellectual achievements, but whether he ever existed. We do know that a group of people who said they were followers of his - the Pythagoreans - emerged around the fifth century BC. Melvyn Bragg and his guests discuss what we do and don't know about this legendary figure and his followers, and explore the ideas associated with them. Some Pythagoreans, such as Philolaus and Archytas, were major mathematical figures in their own right. The central Pythagorean idea was that number had the capacity to explain the truths of the world. This was as much a mystical belief as a mathematical one, encompassing numerological notions about the 'character' of specific numbers. Moreover, the Pythagoreans lived in accordance with a bizarre code which dictated everything from what they could eat to how they should wash. Nonetheless, Pythagorean ideas, centred on their theory of number, have had a profound impact on Western science and philosophy, from Plato through astronomers like Copernicus to the present day.Serafina Cuomo is Reader in Roman History at Birkbeck College, University of London; John O'Connor is Senior Lecturer in Mathematics at the University of Saint Andrews; Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick.

Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash Radio 4. I hope you enjoy the program. Hello, the ancient Greek mathematician Pythagoras is probably better known than most of his illustrious successes over the last two and a half thousand years. This is thanks in part to the theorem concerning right-angled triangles that bears his name. Yet he left no texts behind. We know next to nothing about his life. Some scholars have doubted that he ever existed. We do know that a sect called

the Pythagorean's emerge in Italy in the 5th century BC, but they're as well known for their bizarre beliefs as for their mathematical innovations. Nonetheless, the ideas associated with Pythagoras and the Pythagorean have had a deep impact on Western science and philosophy, inspiring Plato and Euclid,

Copernicus and Newton. At the core of this is their belief that the truths that underlie reality can be found through numbers. With me to discuss Pythagoras and Pythagoreans are Serafina Cuomo, reader in Roman history at Birkbeck College, University of London, John O'Connor, senior lecturer in mathematics at the University of St. Andrews, and Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick.

Ian Stewart, how advanced was mathematics in the 6th century BC? Not a lot. It depends on which area you look at. The whole of the Middle East had quite a strong mathematical tradition. and the Babylonians a thousand years ago knew quite a lot. I mean a thousand years before? A thousand years before, yeah, a thousand years before Pythagoras, so 1500 BC. The Babylonians really knew a lot of mathematics in various ways. They could solve cubic equations, for example.

What else? Quadratic equations, particularly they knew about the square root of two. Their method for cubics was rather rudimentary, but they nonetheless had one. So they knew, I think, more than the population. Pythagorean's knew, because at the time of Pythagoras, that particular cult seemed to have gone back to rather lower-level mathematics in many ways. But the Pythagorean's knew about triangular numbers, 1 plus 2, plus 3 plus 4, that kind of thing. They knew about the square root of 2 being an irrational number, not being a fraction, and they knew Pythagoras' theorem about the sides of a right of. triangle. Can we just

give us some idea of the swirl of ideas around that time? It's an awful long time ago, and we're going on an awful long time period, but it is fascinating because the idea of sects and segments of society doesn't seem to apply. People are getting ideas from here, there, and everywhere. It's drifting out. Can you give us some idea? The Babylonians, we could go back to Egyptians, we could bring in greater Greece, and just fill it out a bit.

The Babylonians were very strong on astronomy, and I think quite a bit of their mathematics was for astronomical purposes. They had very good observations of the movements of the planets. There are surviving clay tablets with cuneiform inscriptions which can be recognised as tables

of the movements of Jupiter and so on. They also use mathematics for more practical purposes, taxation, land measurement, that kind of thing. The Egyptians in some ways were not as advanced as the Babylonians, I think the Egyptians were focused a little more on the practical side of things.

For example, they knew an awful lot about the mathematics of pyramids, as you might expect. And the whole culture is a huge mixture, and people are perpetually invading other people. But the ideas are floating around. And Pythagoras, it is said, went to Egypt and picked up some Egyptian ideas, although a lot of scholars say we haven't got a clue whether he really did. But what about, just before I move on, And what are the most important ideas associated with Pythagoras and the Pythagoras? Just a brief resume maybe.

From a modern point of view, the most important one is the idea that number is the basis of nature, that the physical world works on numerical, mathematical principles. And some of their evidence for that came from musical sounds, and some of it was pure numerology. They understood some very basic facts in geometry such as Pythagoras' theorem They understood that irrational numbers existed

On the other hand they embedded this in a kind of mysticism and numerology And the Pythagorean's were a religious cult More than anything So it's all mixed up But there's a lot of interesting stuff Sort of in there among the mysticism It's interesting that the start of this is as much mystical and religious as mathematical.

But Seraphinaquem, let's get to the man himself. What's the traditional story of Pythagoras until, say, the end of the 19th century? Who was he thought to be? The traditional stories see him as a kind of holy name as well as a philosopher, as a traveler who traveled to Egypt, perhaps Mesopotamia, but also to the underworld and back. So the traditional stories painting, as almost a supernatural figure.

According to some of the anecdotes about him, he had a golden thigh that he displayed to the public in Olympia. He could speak to animals, including eagles and bears. He could predict earthquakes. He could remember his previous lives. So the traditional picture does the people, to us something about what we would call scientific

achievements, but the side of him that was about spirituality, religion, miracles even, is much stronger. But that's common with people who found all sorts of cults, isn't it, around that period, they're accredited with great miracles

being in two places at once, going to the underworld and coming back, that transcript over the centuries, we then come to Christianity so that isn't unusual. But there is a an inside story that he was born in Samos, one of the Greek islands, which was it was a pre-Socratic society.

It was half Persian, half Greek, which made it intellectually vibrant and so on. A story was tracked out as this is what this man did as a man. Yes, we seem to know for sure that he came from Samos, which is off the coast of Asia Minor. So at what we could call the interface

between east and west, the boundary, if you like, between the Greek world and the Persian world. For reasons that are not well, known but could have to do with his involvement in the political life of Samus. He emigrated and ended up in Croton in southern Italy where he seems to have made contact with the local authorities who encouraged him to teach young people of Croton and the result is the foundation if you like of the first so-called Pythagorean sect.

Now current scholarship or some current scholarship would say that none of this obtains that thoroughly investigated, nothing stands up, that he perhaps didn't exist at all. What's your assessment of the current scholastic view of the existence or not of Pythagoras? Yes, for a very long time, scholars were more intent in trying to shift truth from reality. The major contribution to this operation came in 1962

by a scholar called Walter Burkert. The book was translated into English. in 1972. Anyway, Burkert, through a very careful and thorough analysis of the sources, demonstrated, for many people, demonstrated in an uncontrovertible way, that Pythagoras was what he called a shaman, a religious figure, and that he made little, hardly any contribution to what we would call science.

After Burkett for many times, most scholars would just agree with his view. In more recent years, the end of the 90s and 9,000, the notice and so on, more work has been published that is trying to rehabilitate some of the scientific side of the Pythagorean's. And actually said that there might have been a man called Pythagoras. That is a bit of an open question still. It is actually known as the Pythagorean question.

Yes. It's another programme. It's a power of oral history, which we miss out on this program a lot. because we have a cult of written records and oral history lasted over several centuries in certain civilizations. Anyway, that's another question. John O'Connor, we know a lot more about the Pythagoreans.

So did they think they were following somebody who actually existed or did they make the word up? I think they did think that they were following somebody that existed. I've always liked to think of Pythagoras as actually a concrete character who was the leader of a religious, cult. But some of the people who came later and actually followed on Pythagoras as a chap called Philo Laos who was the first person to actually publish anything about the Pythagorean's

doctrine. That Pythagorean's were a secret cult. They didn't publish what they knew, what they discovered. And that's one of the reasons that the written record is so difficult to follow, that it wasn't until quite a lot later that people actually started saying what things the Pythagoras had done, and in particular in the case of Philalows, he published a book called On Nature, of which only fragments remain. And Burk had actually reinstated some of the things that Philalows would allege to have done. For many years it was thought that most of the Philalows' work was plagiarized, was made up by later people, because Pythagoras had such a following in the centuries after he died that the Pythagorean legend actually expanded greatly.

And so most of the sources that we have for Pythagoras were written hundreds and hundreds of years after. He actually did anything that people claimed. The word religious has been used several times, and you've used the word cult as a sect, as a secret sect, which makes it very difficult for us to know enough about it. him they don't seem to have kept records. What was cultish about them?

And perhaps you can bring in the fact that the unusualness is women came into this. That's right. Hypatia of Alexandria. I just wanted to say the name, actually. She was part of it as well. Yeah, she was a rather late part of it. But the cult was, it was a secret cult in the sense that they were people who lived together as a group. They lived with quite an austere life, it's claimed, that there are some of the latest stories

are that these people didn't wash very often and so this actually kept them separate from other people. They had a lot of rather strange practices. One of the most best known is about eating beans I don't particularly want to go into but they had all sorts of other prescriptions as well that if you took your shoes off you were supposed to take the right shoe off first.

If you washed your feet, you were supposed to wash your left foot first. There must have been more to it than that though. I mean, what were they believing in? I mean, you can't believe in. taking your shoes up. Well, maybe you can. Well, these were some of the things I think would help to bind the cult together. The beliefs of the cult, though, were

actually rather more interesting. They believed, for example, in the transmutation of souls. This is something that they allegedly got from Pythagoras. They believed, in a way, in a way that influenced a lot of later philosophers that souls and bodies were actually separate.

That the soul was something which, actually could live on after the body was dead and could actually move on into other bodies, even into other animals. And there are stories told about how Pysagos recognized one of his friends in the barking of a dog. But this idea of the cell played through for the next hundreds of years and then gathered great strength,

particularly in Christianity, but another thing, this is an important thing. I mean, it's taken up in later Greek thinkers. It's a mile away from putting a best foot forward of every was. That's right. It's something which grateful to influence the platonists afterwards. And before a moment, can we just emphasise that it was open to women, and that made it rather extraordinary?

Well, I don't know that we know very much about the other sort of sex that were available then, but certainly it was open to women, and the fact that these people lived together in, if you like, quite an austere way as a mixed group was actually something I think that is quite interesting, as early as that, and certainly later on, other groups of intellectuals that live together in monasteries or convents, and they were usually segregated. So the Pythagoras, the Pythagoreans, were actually quite in advance of their time from that point.

And can you just briefly, there's a couple of his pupils whom we do know enough about to say they unequivocally exist? I don't know about unequivocally, but... Nobody unequivocally exists. Well, none of us do, anyway. There was somebody called he Parsis, who was one of the people early in the cult. and there are all sorts of stories about him that he was the person who possibly discovered that the

square root of two was a number which couldn't be written as a quotient of integers. He may also have discovered how to construct a dodecahedron and there are stories that because of revealing this to other people

the gods in their wisdom actually drowned him. So there are stories like that. Those seem rather less well documented than some of the stories about Philo Laos. Philo Laos is somebody that did go on to actually do some further mathematics and is somebody who at least some of what he did was actually written down and so Aristotle actually refers to some of the things that he did.

And Architas is referred to by Plato isn't he? Yes, that's right. And taught Plato? Arquetus is, well it was a friend of Plato that the the thing that's best known about him in all of his early biographies is that when Plato was foolish enough to go and visit a place called Syracuse

which is just around the corner from where these people were hanging out in southern Italy in Sicily he was detained by the person who was called a tyrant of Syracuse. Tyrants were well the word is now regarded more pejoratively than it was then

but he was detained by Dionysius and Architos was sufficiently powerful in southern Italy that he was actually able to send a ship to rescue Plato and that's the thing that people knew most about Arquetus in antiquity. So we have lines developing, even though they might be fragile and open to dispute and non-existent, perhaps, who knows? It's a mysterious part of intellectual history.

Can you, Ian Stuart, said that you think that their insight into the truth of reality being interpreted best through number was very powerful, long-lasting and an astonishing insight. Now, can you

talk to that? This idea, I think, has formed a lot of modern science. If you trace the history of modern science and you look at the pivotal points, you repeatedly find great scientists, Mike,

let's say Kepler, who are looking for mathematical patterns in nature. They start out with the idea that they're not even quite sure what the question is, but the answer is a mathematical answer. So Kepler looked for

the mathematical structure of the movements of the planets. And he published a whole pile of stuff, some of it very speculative. He related the orbits of the known planets to the five regular solids in a way that is now considered to be total and complete nonsense.

But he also came up with three laws of planetary motion, which were used later by Isaac Newton, as the basis for his law of gravity. So this kind of Pythagorean view that numbers are the key to the mysteries of nature has paid off. And even nowadays, we tend to measure how advanced a science is, how solid are sciences, by how mathematical it is.

How important were ratios and fractions? This is going to stir the... Still the little graces. That ratios and fraction. Well, let's go back to this chap, hipposus of metapontum and the whole business of the square root of two not being an exact fraction. The idea of number prior to that discovery

was essentially whole numbers, things you can count, one, two, three, four. But if you've got things like one, two, three, four, you rapidly come to the feet. that a half or three quarters or fractions of this kind are also perfectly reasonable numerical quantities. They're what happens when you split one number into a lot of equal pieces. But it's all built from whole numbers. So the Pythagorean's have this idea of everything in the universe being built from number,

and to them this means whole numbers and by extension fractions. On the one hand. On the other hand, they have Pythagoras' theorem about right triangles. which tells you in modern language that if you have a square with side 1, then its diagonal has length root 2, the square root of 2. And those two fundamental things that the Pythagorean's believed, and were convinced in, contradict each other, because the square root of 2 is not a fraction.

So this means that geometrically you can start from perfectly reasonable things like whole number. numbers a square of side one. And out of this emerges a natural object which must exist, it's diagonal, and you know that that's not one of the numbers that you're happy with. But it worked, and it was solved, and we

will come to this letter in the programme. But, Serafina, let's take a number. I'm trying to keep the two things going, although mathematical and the mystical, as it were, because numbers had all sorts of meanings for them, the number 5, the number 28, the number women were, even numbers,

Manua? Can you just talk to that? Yes, I think the first thing to keep in mind is that we talk about the mathematical and the mystical as if they were two separate categories. Because in our society, science and religion, spiritual and scientific are two very different categories. But we know through the history of science that up until at least Newton, the two categories were not seen as entirely separate. They meshed into each other. So for them it made perfect sense that you can understand reality through number.

But number means both things to do with the square root of two and things to do with what two means. So two may mean union. Five means marriage because it's the union of the first odd number and the first even number. One doesn't count as a number, does it really? One doesn't really count as a number. So we've got two and three are the first number. Yes. One is not odd and is not even in ancient mathematics in general, and for the Pythagorean's in particular.

The status and nature of one is one of the most debated topics in ancient mathematics. So let's just forget about one. We did a program on it. Seven was seen both as the number of opportunity because children can come out of their mothers, fully developed at seven months because adults are developed at seven by two, which is 14. But seven is also the number of Athena,

the Virgin goddess, because seven doesn't generate anything and it's not generating by anything. So I think the two things were really in parallel. What we consider scientific and what we consider numerology, for them were one and the same thing. So that's why five,

was marriage because two plus three equals five, the man and woman equals marriage. And they found perfect numbers like 28, which the divisors would add up to 28 and so on. Yes. Yes. Any more? I think five could also be justice. Because it's a kind of, the union also means that you found a kind of harmony and balance between two opposite principles. and justice could be seen as fairness and balance. John O'Connor, one famous aspect of the everyday world that could be explained to some extent, maybe to a huge extent by number, was music, which is attributed to Pythagoras and the Pythagoreans.

Can you say it a bit about that? Yes, one of the things that I still tell our first-year undergraduates when they first meet the thing called the Harmonic Series, which is a series, which goes one plus a half, plus a third, plus a quarter and so on, is that the reason it's called harmonic is because of Pythagoras, that the story goes that Pythagoras was passing a forge one time and he heard the smiths banging away their hammers on the anvils.

And so being a curious fellow, he went in and realized that sometimes the notes when they bang two things together sounded harmonious and sometimes they didn't. And so he went in there and they measured the things that they were striking. and they discovered that the harmonious notes came from when you had two things which were a size which was represented by a small number. So, for example, if you had one thing which was twice the size of another, then it sounded harmonious, whereas if you had one thing that was, say,

seven-eighths the size of another, it didn't sound harmonious when they were struck together. That's how the story goes, and that's the reason that mathematicians still call a harmonic series harmonic. It's more likely, I think, that the piebales, Pythagoreans came upon this knowledge by watching people tuning musical instruments, because that is one of the places where you do have to worry about that. And in particular, Philalows and Architas actually developed a very good theory, very similar to the theory that we have now, of how notes match up.

So that, for example, if you halve the length of a string, then the note goes up by one octave. So the ratio of 2 to 1, 1 to 2, is an octave. If you take a note, which is you reduced a string by two-thirds, then the note goes up by what we now call a fifth. Reduce it by three-quarters, then it goes up by what we now call a fourth. And Arquitos had worked out that if you took the two ratios corresponding, that's three over two and three over four, and you multiply them together,

then what you ended up with is you ended up with two. So that if you take a fifth and a fourth and put them together, then you get a whole octave. and if you do something even cleverer and you go up by fifth and down by fourth then you get a ratio of 9 over 8 and this is what Arquita's called the tone and that's what we would now call two semi-tones

because it's actually two notes on the piano and one of the reasons that Arquitas did much of the mathematics that he did was to try and work out if you could actually make a genuine semi-tone that's to say if you could actually split the tone and the 9-8s ratio into two other ratios which would actually form something in between those two.

And he was able to prove that you couldn't. And that's actually one of the sort of the first applications of some really quite advanced number theory to this particular idea. What it means is that by great good fortune, if you take this ratio of 9 over 8 and you take its sixth power, you get almost 2.02, which means that you can fit almost exactly six tones into, an octave. And that's the basis

for the modern piano, the fact that you can actually very nearly and actually squeeze these things. You can temper the ratios a little bit, as we now call it, and fit these six tones into a whole octave. And so if they'd had the capability, they could have built a piano back in those days, but of course they didn't have the technology to actually

do that. Ian Stewart, let's turn to this right-angle triangle, which delighted us all so much. And the hypotenuse, one of the nice as to words that we learned, doesn't it? Right. Why is it so important? It's important for the whole of mathematics, in fact, because

what it does is it relates three lengths. It relates the two shorter sides of a right angle, triangle, and the longest side, the hypotenuse. Probably the only time in your life you ever get to use the word hypotenuse is at school

when you're talking about these things. So, for example, in later mathematical developments when Descartes, introduced so-called coordinate geometry, where everything's in terms of every point in the plane is given by two numbers, how far east or west it is, how far north or south it is, and suppose you say, well, let's go three units east and then four units north, how far am I from where I started? Those are two sides of a right-angled triangle. And Pythagoras' theorem tells us that we can calculate how far away we have gone by squaring those two

numbers, adding them up and taking the square route, which in this case happens to give you five. So if you know distances horizontally and distances vertically, you know all distances. So geometry can be reduced to number and the concept of distance is

built in thanks to Pythagoras' theorem. So this is one of the ways that it influences the subsequent development of mathematics because it helped to fuse the world of numbers and the world of geometry together into two different aspects of essentially the same thing. Sada Fina Cuomo, is there a sense in which, again, ascribing would that be,

is this idea the right-angled triangle was around at the same time or even before Pythagoras is said to have been around? We find it in Egyptian, mathematical culture and Babylonian and so on. Yeah, we find it in other cultures. We find, for instance, in Mesopotamian mathematics, knowledge of so-called Pythagorean triplets, a series of numbers that correspond to the sides of a right-angle triangle, 34-5 would be the first Pythagorean triplet. What we don't seem to find is a demonstration, a proof of the theorem, in an axiomatic or deductive structure, such as we find in uglitz elements, which is the first extant proof of Pythagoras'

Pythagoras' theorem. So the knowledge was there, but the proof came only later. So it's a question, it's an interesting example of technology, getting on with using this, to build things and do things, not feeling necessary to prove it until, or maybe

not being able to, which was it? Were they not able to prove this theorem? Or did they just not think it was necessary? That's a very controversial question. Myself, I think that actually they weren't interested in proving it, because if I had to say they weren't able, I'd have to make assumptions about mathematical ability

of different civilizations that I don't think we can make, really. I think mathematics, like everything else, is driven by some kind of need as well as by some kind of curiosity. And I think just different groups of people at different times have been driven by different curiosities and different. needs. That's a wonderful sidestep.

But Ian put his finger up, so let's see what happens. Well, I just want to reinforce what Seraphina said, but say, I mean, the whole concept of waterproof is has changed over the years, and I think Euclid is the first place where we know that someone seriously sat down and wrote down

some idea of what it really means to give a logical proof of something. But this idea that somehow you must have evidence that what you're saying is true must go back further. I mean, we can reconstruct, for example, how the Babylonians solve quadratic equations, and it's pretty clear that they knew how to do it and they knew it was right. And in the case of the Pythagoras' theorem, you can draw fairly simple geometrical diagrams

and point to them and say, look, it's obvious. So if this was on television, we could show a picture which would convince everybody in the audience immediately that Pythagoras' theorem must be true. So I think they may have known this kind of slightly experimental but very convincing argument. Ian mentioned earlier about hitting a problem with belief that whole numbers are things that matter, John O'Connor. Can you talk to that the problem they hit?

Yes, the idea of number is actually really quite a complicated thing. that the idea that if you've got five apples or five inches, then these are examples of the number five, it's actually quite a sophisticated idea. The way I think the Pythagoreans got round that was the use of the monad. Why is it a sophisticated idea? Because most people are listening to say,

it's that obvious, I've got five fingers, I'm holding, well, I haven't four fingers in the thumb, and that isn't a sophisticated. Well, that means if you're comparing this to apples, then you have to actually make a correspondence between one apple, two apples, three apples, and so on with your fingers. The idea of the number really comes from taking,

if you like, the ratio of five apples to one apple. That gives you the number five. If you take five inches to one inch, then you get the ratio of five. So really an abstract number is really always a ratio. And it's a ratio by this magic number one, the monad, the other thing that generates all of the numbers. And that's one of the reasons why one was thought to be different

from all of the other numbers. It's the thing that you always divided by, whereas the other numbers were the things which you actually achieved as ratios. And so the idea of number as coming in this sort of way, as a number in particular as coming as ratios, is something that I think goes all the way back to the Pythagoreans, the idea that you've got fractions, that those are the ways that you actually build up numbers.

And that's the way that numbers really came in as abstract quantities rather than as particular examples of five apples, five fingers, five inches. Saraphena Cuomo, let's begin to talk about the influence now from the fourth century BC. How much influence did the Pythagorean have, say, on Plato and his followers? What sort of influence? The influence on Plato and his followers is enormous. In fact, in late antiquities, some Neo-Pythagorean would go as far as saying that Plato was just a Pythagorean. I would pick out two main areas of influence on Plato and Platonism.

One is to do with beliefs in the soul. So the distinction between body and soul and the fact that the body perishes while the soul continues in one form or the other is something that Plato was very keen on. It's discussed in the dialogue where he talks about Socrates's death. and two of the characters in the dialogue are actually Pythagorean's, and they talk about how Socrates's death is not the end of everything because he's so going to live on. And the second main area where Plato could be said to be influenced by Pythagoras and the Pythagorean is, of course, the fact that number can provide a key

to explain reality. In his cosmogonical treat is called the Timetian, Plato, through his characters, tells how the universe began. And you soon find out that the universe is organized along mathematical principles. All five elements have a geometrical structure, the harmony of the universe, the correspondences between souls and bodies are governed by mathematical proportions. So in these two areas, Pythagoras influenced Platonism, and then because of the major influence of Platonism on later thinkers,

that's how Pythagoras' thoughts lived on. John McDonagoras, the Pythagoras, we've mentioned him before, had a relationship with Plato, seems to save his life, which is quite a strong relationship. Did he lay the ground for the idea of reasoning from basic truths, which broadly Plato developed? I don't think so. Arquetus was certainly not...

I mean, he was too early to be able to prove things, if you like, from axioms. He didn't have an axiom system to set up in order to prove things. He did, however, actually do some, what we would now think of as constructions, that he actually did some rather clever things. He actually managed to produce something which had been, looked for for a long time, the cube root of two, the Delian problem of doubling the

size of an altar. He was actually able to make a three-dimensional construction to actually demonstrate that you could actually construct this particular number. Now, you couldn't construct it using the ruler and compass methods that people spent a long time looking for constructions with, but he was able to do this with quite a complicated three-dimensional construction. People accused this of being mechanical, but I mean it wasn't mechanical, it was really just the same as any of the other constructions that were people were using in two dimensions. The platonic idea that what you have is that you have

a sort of virtual world of which the real world is a concrete representation, so that when I draw a triangle on the blackboard and reason from that, I'm really reasoning about a virtual triangle which has the sort of properties that I can actually then deduce from my imperfect diagram. And so Arquitas was able to construct this cube root of two using something which, again, is a sort of a mind experiment, but it was in three dimensions rather than in two. In short, you referred to Euclid once or twice, and in this conversation, as have others. The elements of Euclid are one of the great books in the world,

into laying down the laws of geometry and mathematics. Can you try to tell the listeners what influence you see? It's a murky area, but what influence you see that Pythagorean's having on that, on Euclid, on that work? It's clear if you read Euclid that there must have been an awful lot going on that led up to what Euclid wrote down. It's very unlikely that some towering genius will simply sit down one day and start writing this stuff.

there has to be a long tradition that he is collecting, formalising. And that tradition, among other things, must have its roots in the Pythagorean's and their way of thinking about things. So what Euclid does is really say, let's systematize this, let's build it on first principles, let's strip it right back to the simple ingredients, straight lines, points, circles and things like that. Let's go for what I think Plato had, the sort of, ideal world of Plato.

Let's write down what properties this little mathematical piece of that ideal world will have and then develop the whole of geometry in particular but also number theory a number of other things are actually in Euclid which are often not taught today

the whole of the theory of irrational numbers is in Euclid that goes back to Eudoxus. We have some idea where that comes from and the culmination of Euclid is the classification and construction of the five regular solids, often called the platonic solids, and known to the Pythagorean. Certainly the Dodecahedron was known to the Pythagorean's.

So it's almost as if the objective of Euclid is to put some of this Pythagorean stuff on a sound footing. But I suspect there must be lots of other influences that came in as well. Seraphina Quama, the Pythagoreans, they had an unusual view of the cosmos. Can you talk about that? The view of the cosmos that is unusual is reported in Aristotle and is now attributed to Philoleus. A Pythagorean, yes.

Writing in the second half of the 5th century BC, it was unusual because instead of putting the earth at the center, it posited what they called a central fire. To give a name to the central fire, they called it by the name of the goddess of the east. her hastia. So Aristotle reports this model that is very unusual and for him completely wrong where you have a central fire and all the other heavenly bodies rotating around it. Now Aristotle criticized it. Not many people in antiquities seem to have

picked up on it, but it seems to have inspired Copernicus. The book that created eventually the heliocentric revolution is full of references to the Pythagorean's and in particular to Philoleus. In the dedicatory letter to the

Pope, Copernicus even mentioned by name, Philolaus and some other Pythagorean as inspirations. So they were the model. The text had gone through probably translated into Arabic then back into Latin. So the text had gone through

and they picked up this idea of the central fire, that planets are moving around the central fire, which was that easy to relate, I suppose, to the sun. Yes, Copernicus didn't know the text itself, but there are reports of it in Cicero and Plutarch, which he would have known.

Ian, Ian Stewart, and this drives through, this connection goes through. This goes through, it goes, there's a very strong, very clear link, which goes essentially Copernicus, Galileo, Kepler, Newton. and Isaac Newton is still in this mystical tradition.

He's often thought of as the first of the moderns, but there's a lovely quote from John Maynard Keynes, which says, no, he was the last of the magicians. He spent as much time in alchemy as he did on mathematics and physics. And he thought his greatest work was on the book of Revelation. But, yeah, so all this stuff with calculus and the motion of the planets and everything else, is just a sort of warm-up for the serious work on Revelation.

It's interesting, back to what Serifino was saying earlier in the program, that these two things were inextricably combined, that there wouldn't be the distinctions that we take up so readily. I think even today a lot of scientists at the back of their minds is a little bit of this Pythagorean view that it's got to be mathematical, hasn't it? Why are the physicists looking for a theory of everything? They want an equation you can put on your T-shirt.

They want the mathematical explanation of the world. It's a very Pythagorean thing to want. Finally, and I'm afraid briefly, John O'Connor, do you think it is still around the Pythagorean notion? If you're a physicist working on your latest version of string theory, then you're still actually going back to your Pythagorean roots. You're still believing that there is a mathematical explanation out there which will actually explain anything that nature can throw at you.

Right. Well, having asked you to be brief, you're exemplary brief, and I'm now paddling away. to be in a few more seconds. Well, I thought that was terrific. I hope I can remember what I've learned. Anyway, thank you very much to Serafina Cuomo. There are John O'Connor and to Ian Stewart. We won't be here next week. John Simpson will be here. We'll be back in a fortnight with the samurai. Thanks for listening. If you've enjoyed this BBC podcast, why not try others, such as Material World, where Quentin Cooper discusses everything from archaeology to zoology. To find out more, visit bbc.com.ukho.